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C2×#13103974 #72226572

order3200
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC2×#13103974

Splittings

SylowC52 ⋊ C2×C4×C4⋊C4

Subgroups

subgroupG/H
centerC23C2×D5⋊AGL(1,𝔽5)
commutatorC2×C52C22×C42
FrattiniC22C22×D5⋊AGL(1,𝔽5)
FittingC22×C4×C52C2×C4
socleC23×C52C22×C4
prime radicalC23×C5⋊D5C23
cosocleC22×C5⋊D5C24
hypercenterC22×C4D5⋊AGL(1,𝔽5)
hypocenterC52C2×C4×C4⋊C4
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4

2354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 160

A minimal faithful permutation representation — 4 generators on 20 points:

(3,5,6,4)(12,13,14,15)(16,17,18,19);(1,2)(11,12,15,13,14);(3,4,6,5)(8,10);(3,4,6,5)(7,8)(9,10)(17,18)(19,20)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#72221487
maximal normal subgroup ofC2×#72221487
maximal quotient of#72226573

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,5,6,4)(12,13,14,15)(16,17,18,19), (1,2)(11,12,15,13,14), (3,4,6,5)(8,10), (3,4,6,5)(7,8)(9,10)(17,18)(19,20));